Answer
$4$
Work Step by Step
Step 1: $[2(\cos135^{\circ}+i\sin135^{\circ})][2(\cos225^{\circ}+i\sin225^{\circ}]$
Step 2: Using the product theorem, the expression becomes
$2(2[\cos(135^{\circ}+225^{\circ})+i\sin(135^{\circ}+225^{\circ})]$
Step 3: Simplifying, $4(\cos360^{\circ}+i\sin360^{\circ})$
Step 4: We know that $\cos360^{\circ}=1$ and $\sin360^{\circ}=0$
Step 5: Substituting these values in the expression, $4(1+i(0))$
Step 6: Therefore, the final answer is $4$.